Blok uses only the cookies needed to run the site and remember your preferences. See our Privacy Policy.
An insurer promises money if something bad happens; you pay premiums whether it does or not. Every formula in this course prices that promise.
The possibility that something bad happens; in particular: those events that result in financial loss.
Note what that definition rules out. A bad thing that costs you nothing is not risk in this course. The loss has to be FINANCIAL, because the thing an insurer can hand you is money.
You take steps to mitigate the financial loss involved — for example, buy an insurance product.
The insurer agrees to pay out money — BENEFITS — at specified times, upon the occurrence of specified events causing financial loss.
In return the insured pays PREMIUMS.
The contract between insurer and insured is the insurance POLICY.
Three words, and the whole course hangs off them. Learn which direction each one flows.
BENEFIT — insurer pays you. PREMIUM — you pay the insurer. POLICY — the piece of paper that says when.
Insurance trades uncertainty for certainty.
The risk is then transferred from the individuals facing the loss to the insurer.
Say you would lose 200 000 if you died tomorrow and your family lost your income. That is a huge, unpredictable loss.
Swap it for a premium of, say, 300 a year. Small, and you know the number. That swap IS the product.
Insurance involves a sharing or POOLING of risks among a large group of people — among all those who purchase insurance from a particular company.
"The contributions of the many to cover the misfortunes of the few."
— the motto of Lloyd's of London.
Pooling is why the insurer can survive doing this. It cannot predict WHO dies next year. Across ten thousand policyholders it can predict roughly HOW MANY.
That gap — unpredictable for one, predictable for many — is the entire business.
A business professional who deals with the financial impact of risk and uncertainty. Actuaries provide assessments of financial security systems, with a focus on their complexity, their mathematics, and their mechanisms.
Actuaries mathematically evaluate the probability of events and quantify the contingent outcomes in order to minimize the impacts of financial losses associated with uncertain undesirable events.
What the course puts focus on:
At the same time, the course creates awareness for the societal impact of the technical concepts discussed — for example in discussions related to the sustainability of pension systems, or the affordability of essential life insurance products.
Yes. The definition says benefits are paid "upon the occurrence of specified events". The specified event here is SURVIVAL to a date, not death.
This product has a name — a PURE ENDOWMENT — and it turns up in Chapter 4. Do not assume "insurance" always means "pays on death".
The payment happens only if something about a person's life happens. That one word, contingent, is why probability has to enter the pricing.
The insurance policy pays a benefit CONTINGENT ON (i.e. depends on) the death or survival or the health status of the insured risk.
Contingent means conditional. The payment is not scheduled — it is triggered.
Compare two contracts: • A bank loan repays 1 000 at the end of year 5. Certain. • A life insurance pays 1 000 at the end of the year you die. Uncertain, both in AMOUNT of time and in WHETHER it is within the term.
Pricing the first needs interest only. Pricing the second needs interest AND a model of how long people live.
That is the whole course in one sentence: combine the time value of money with a model of human mortality.
Traditional life insurance contracts include:
Modern life insurance contracts include:
In Belgium and the Netherlands, universal life is sold as "tak 21" and equity-linked as "tak 23".
Life annuity products:
The two families this course values, and the trigger for each.
INSURANCE — one lump sum, triggered by DEATH (or by reaching a maturity date). ANNUITY — a stream of payments, each one triggered by BEING ALIVE on its payment date.
Chapter 4 prices the first family. Chapter 5 prices the second. Chapters 6 and 7 use both to work out what premium to charge and how much money to hold.
An annuity — a stream, each payment contingent on being alive.
What is uncertain is the NUMBER of payments. The amount of each one is known; how many there will be is not.
That is exactly the quantity Chapter 2 calls the curtate future lifetime, .
Term, whole life and endowment insurance. Learn what each one pays and when, because Chapter 4 gives each of them a symbol and never explains the product again.
The three traditional forms of life insurance are TERM, WHOLE LIFE and ENDOWMENT insurance.
These policies dominated insurance markets until the 1980s, and in some countries are still popular today.
Term (or temporary) insurance pays a lump sum benefit on the death of the policyholder, PROVIDED death occurs before the end of a specified term. Typical contract terms range from 10 to 30 years.
The premiums for term insurance are usually very small relative to the sum insured, because the insurer has to pay a death benefit on only a small proportion of the policies issued.
A policyholder aged 40 buys 10-year term insurance. Roughly what fraction of such policies ever pay out?
The insurer pays only if the life dies before age 50.
That probability might be around 2%.
So around 98% of such policies expire with no death benefit payable.
About 2%. The premiums from the other 98% subsidize the benefits for the 2% for which the death benefit must be paid.
That subsidy is pooling again, seen from the insurer's books. Nobody who paid and did not die was cheated: they bought the promise, and the promise was kept.
The main purpose of term insurance is FAMILY PROTECTION. For a relatively low monthly cost it protects the policyholder's spouse and children against financial hardship in the event of the policyholder's death.
It is also used to protect businesses against losses arising from the death of key employees — now usually called COLI, Company Owned Life Insurance.
Variants of term insurance:
The word to notice in renewable and convertible term is INSURABILITY. Renewal does not depend at all on the policyholder's health status at that time.
That is valuable precisely because health deteriorates, and Lecture 4's material on SELECTION is the mathematics of that fact.
Whole life, or permanent, insurance pays a lump sum benefit on the death of the policyholder WHENEVER it occurs.
In general, whole life insurance is significantly more expensive than term insurance, relative to the death benefit, as the probability of paying the death benefit (ignoring lapses) is 100%.
Read that again: the insurer WILL pay. The only question is when.
So the price of whole life is not really a bet on whether you die. It is a bet on how long the insurer gets to invest your premiums first — which is why the interest rate matters so much more here than on a 10-year term policy.
An endowment insurance pays a lump sum on death within a fixed term, OR on survival to the end of that term. Either way, the sum insured is paid.
The three products, told apart by one question: what has to happen for you to get paid?
TERM — you must die, AND die within n years. WHOLE LIFE — you must die. That is all. ENDOWMENT — you must either die within n years or be alive at n. Something always happens, so it always pays.
Hold on to the endowment one. In Chapter 4 the endowment insurance is built as a TERM insurance plus a PURE ENDOWMENT, and the reason that works is exactly the "either/or" above: the two pieces cover every possible outcome, and they never overlap.
Because the risk being covered SHRINKS. Each year the borrower repays part of the loan, so the outstanding balance the lender could lose falls.
A decreasing benefit tracks that balance. Level term would over-insure in the later years, and the borrower would be paying for cover they do not need.
This is Tutorial 1, question 1.
Insurance pays once, on death. An annuity pays over and over, for as long as you are alive. Chapter 5 values these, and it is where pensions live.
Annuity contracts offer a regular series of payments. When an annuity depends on the survival of the recipient, it is called a LIFE ANNUITY. The recipient is called an ANNUITANT.
Annuities are often purchased by older lives to provide income in retirement. Buying a whole life annuity guarantees that the income will not run out before the annuitant dies.
That last sentence is the point of the product. The risk being insured here is NOT dying early — it is living a very long time and running out of money.
Insurance and annuities are opposite bets on the same random variable.
Annuities CANNOT be surrendered; there is no cash value once the annuity payments commence.
The main reason is that allowing surrenders would create unmanageable risk of ADVERSE SELECTION — the lives who are most unwell are most likely to surrender.
Work through why that is fatal. If the sick cash out and the healthy stay, the pool left behind lives longer than the price assumed. The annuities the insurer still has to pay cost more than it collected.
Annuity pricing assumes that on an annuitant's death, any excess funds built up from investing the premiums are then used to offset the costs of annuities for SURVIVING annuitants.
Chapter 5 gives this a name — SURVIVORSHIP EARNINGS — and draws it as a box the annuitants pay into and draw out of. Money left by those who died early is what funds those who live long.
Types of annuity contract:
If the policyholder of a deferred annuity dies soon after the annuity commences, there may be some minimum payment period, called the GUARANTEE PERIOD, and the balance would be paid to the policyholder's estate.
Hold on to "guarantee period". Chapter 5 values exactly that product under the name GUARANTEED ANNUITY, and its present value splits into a part that is certain and a part that is life contingent.
Andrew is retired. He has no pension, but has capital of 500 000. He is considering: (a) an annuity paying a level amount for life; (b) an annuity that increases with the cost of living, for life; (c) a 20-year annuity-certain; (d) invest the capital and live on the interest income; (e) invest the capital and draw 40 000 per year. What are the advantages and disadvantages of each?
This is a words question, and words questions on this course are scored on whether you name the RISK each option leaves Andrew holding. So go through them asking one thing each time: what could go wrong, and who is carrying it?
There are only three risks in play — outliving the money, inflation eating the money, and investment returns disappointing. Every option is some combination of who bears which.
(a) LEVEL LIFE ANNUITY. The insurer carries the longevity risk — Andrew cannot outlive the income. But the amount is fixed, so INFLATION erodes it, and he has no access to the capital.
(b) INDEXED LIFE ANNUITY. Removes the inflation risk as well, so the insurer now carries both. The price of that is a LOWER starting payment for the same 500 000.
(c) 20-YEAR ANNUITY-CERTAIN. Pays whether or not he is alive, so his estate is protected if he dies early. But it STOPS at 20 years. If Andrew is alive at 87, the income ends and he has nothing. He carries the longevity risk himself.
(d) LIVE ON THE INTEREST. The capital survives intact for his heirs, and he can access it. But the income depends on interest rates, which vary, and may be small. He carries investment risk and inflation risk.
(e) DRAW 40 000 A YEAR. 500 000 at 40 000 a year lasts around 12-13 years before investment returns are counted. Highest income of the five, most flexible — and the fund can run out entirely while Andrew is still alive.
Credit is for identifying, per option: who bears longevity risk, whether the real value is protected against inflation, whether the capital remains accessible or bequeathable, and whether the income can stop while he is alive.
Early policies lasted one year, so timing barely mattered. Once contracts ran thirty years, the interest earned on premiums became the main event.
A bit of history.
With one-year contracts the time value of money is not a critical aspect. But with, say, a 30-year contract this becomes an important part of the modelling and managing of risk.
Hence, actuarial techniques had to develop beyond the year-to-year modelling of mortality probabilities.
Here is the arithmetic behind that sentence. At 6% a year, 1 EUR promised in one year is worth about 0.943 today. Promised in thirty years, it is worth about 0.174.
One year: a small correction. Thirty years: the payment has lost more than four fifths of its value, and ignoring that would be nonsense.
So a level premium contract has a second, hidden job. Early on you overpay relative to your risk; later you underpay. The difference sits invested, earning interest, and the interest is what makes the later years affordable.
Life insurance works by combining premiums with the INVESTMENT INCOME earned by investing them, such that, by the time the policyholder dies, the premiums plus the investment income are sufficient, ON AVERAGE, to pay the sum insured.
For short-term policies, premiums cover most of the sum insured. For long-term contracts the investment income becomes a much more significant component.
"On average" is doing real work in that sentence. No individual policy breaks even. The pool does.
Chapter 6 turns "on average" into an equation: set the premium so that the expected loss at issue is zero. That is called the EQUIVALENCE PRINCIPLE.
Predicting investment returns over very long terms is very difficult, so insurers tend to calculate premiums using very conservative — that is, low side — assumptions about investment returns.
An insurer can get 7% a year on a 20-year investment, but prices the policy assuming 6%. What is the 1% called, and what does it buy?
The difference between the rate available and the rate assumed is the INTEREST SPREAD.
It is deliberately left in the price rather than passed on.
It is the interest spread, and it covers profit and allows a margin for adverse experience. The risk it guards against is that rates fall below 6% at some point during the contract.
The two ingredients, and which chapter supplies each.
TIME VALUE OF MONEY — how much a future payment is worth today. One number: . MORTALITY — how likely that payment is to happen at all. Chapters 2 and 3.
Multiply them together and sum over every date. That is Chapter 4.
Because of how long the money sits. The 1-year policy pays within twelve months, so discounting changes the answer by a few percent.
The whole life policy might pay in fifty years. At 6%, 1 EUR in fifty years is worth about 0.054 today — so almost the whole price is a statement about interest, not about mortality.
Closed book, three hours, five questions of equal weight, with a formula sheet and a life table you bring yourself. Know the two documents you are allowed before you revise.
Your grade:
Resit: results obtained on assignments no longer count. The grade is determined on the resit exam only.
You may use during the midterm and the final exam: the FORMULA SHEET and the ILLUSTRATIVE LIFE TABLE.
Print your own copy and bring it to the exam room WITHOUT ANNOTATIONS.
Two consequences, and students lose marks to both.
First: you do not have to memorise the formulas. You DO have to be able to find one on a nine-page sheet under time pressure, and to know which of four similar-looking lines is the one you want.
Second: you cannot write notes on it. A worked example scribbled in the margin will get the sheet confiscated.
Revise WITH the formula sheet open, from day one. Not as a crutch — as a map.
By the exam you should be able to answer, in under five seconds: "Which page is the endowment insurance on?" "Which page has and ?" That speed is worth more than any formula you memorised.
The exam is three hours. Each question gets equal weight, unless indicated otherwise.
The papers in the folder all have FIVE questions. Three hours over five equally weighted questions is 36 minutes each. If you are 15 minutes into question 1, move.
The shape repeats year to year, and it is worth knowing before you revise:
Course material you work with:
Teaching activities:
The old-exam discussion in the tutorials is the one thing on this course that is not recorded.
Both the lectures and the prerecorded tutorial exercises are available afterwards. The in-depth exam-question discussion in the tutorial is explicitly not.
Given that the paper reuses its own question shapes year on year, that is the session to be in the room for.
After following this course the student:
The last one is not filler. Most of what feels hard about this course early on is not the probability — it is reading and knowing instantly what is being paid, to whom, how often, and for how long.
Every topic from here on names its notation the moment it appears.
Saying "(a) is safe" earns nothing. Saying "(a) transfers longevity risk to the insurer, at the cost of inflation protection and access to capital" earns the mark.
If the premiums were monthly instead, the first is at and the final one at years.